There are many definitions of "planes", some of them given above.
From personal experience, I would say that if you refer to "astral planes and realms" as other places you can go when out of body, they are more or less endless, and you can create your own as well so that just adds up on the "endless" factor. And so does other people and beings so it seems it's actually just increasing.
With that said, we seem to be able to divide the different realms into different planes, in other words each plane has amounts of sub-planes.
When it comes to physical planes, there seem to be a huge amount of parallel physical planes, or alternate planes, with both slight and major differences depending on how "off scale" you go from where you are now.
And each of these physical planes have various physical planets in an enormous amounts of physical galaxies and solar systems, planets with other forms and even similar forms of life. I'm not sure if you would count these as other "realms" as well though.
Regarding the higher planes, such as the so called "astral plane", it too has many other parallel planes, as well as many different sub-planes within itself, in a sense just like the physical universe has many other planets. All with different types of inhabitants and scenery etc. These lower astral planes are physical in nature, with that I mean the inhabitants have bodies and look physical, and they inhabit worlds that seem physical, such as having nature and other physical attributes, though often not as limited by physical laws as here, such as gravitation. But this also seem to differ from realm to realm, and from inhabitants to inhabitants. One example is that I may be able to fly and do various "tricks" in one world, and the inhabitants there may not be able to do so and are amazed.
You also have "spirit worlds" where people whom are diseased are now living. Most of these also seem to be "lower planes" that are still close to the physical plane and thus a sort of reflection of the physical, with the inhabitants having a physical appearance, their worlds looking physical etc, though magical powers and "fantasy movie" kind of things are not so unusual there.
These lower astral planes also seem to be somehow connected to the physical plane / world, and even though it may seem like heaven, and many of those who live their think they are in heaven (or hell for that matter, for those more unlucky ones) in reality they are just in lower planes no matter how beautiful they seem. The higher astral planes are less physical looking in nature and the higher you get the less physical the appearance there is to the worlds and their inhabitants, such as the "Buddhaic plane" as mentioned above.
People in general seem to be trapped in the lower astral planes after death due to their obsession with the physical and material, some are not even aware of being dead and keep on living their lives there as they did here, in worlds identical to this one more or less.
I've found that even going out of the physical world and into space can be somewhat troublesome, as when you project upwards towards the sky you'll find yourself in other slightly higher astral planes, and you may have to fly up through several of these before even reaching out in just the physical space outside our Earth.
From there you can propell yourself further upwards if you'd like while having in mind to experience the higher planes and true reality and your highest self, even a short affirmation said out loud while in the astral planes such as "higher self now" can be enough to have you lifted up towards the highest experience far beyond the lower planes.
Personally I refer to all planes above the physical as "astral planes" for simplicity , I just call them higher and lower astral planes of different degrees.
So the bottom line is that your question is rather complex, and the answer is that there seem to be infinite amounts of "planes and realms"

How many "major" planes there are before the last and ultimate one, I personally have no idea.
-EA